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Discussion on some coupled fixed point theorems
Fixed Point Theory and Applications volume 2013, Article number: 50 (2013)
Abstract
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature.
MSC: 47H10, 54H25.
1 Introduction
In recent years, there has been recent interest in establishing fixed point theorems on ordered metric spaces with a contractivity condition which holds for all points that are related by partial ordering.
In [1], Ran and Reurings established the following fixed point theorem that extends the Banach contraction principle to the setting of ordered metric spaces.
Theorem 1.1 (Ran and Reurings [1])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
T is continuous nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
there exists a constant such that for all with ,
Then T has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
Nieto and López [2] extended the above result for a mapping T not necessarily continuous by assuming an additional hypothesis on .
Theorem 1.2 (Nieto and López [2])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
if a nondecreasing sequence in X converges to some point , then for all n;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists a constant such that for all with ,
Then T has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
Theorems 1.1 and 1.2 are extended and generalized by many authors. Before presenting some of theses results, we need to introduce some functional sets.
Denote by Φ the set of functions satisfying the following conditions:
() φ is continuous nondecreasing;
() .
Denote by the set of functions satisfying the following condition:
Denote by Θ the set of functions which satisfy the condition:
Denote by Ψ the set of functions satisfying the following conditions:
() for all ;
() .
In [3], Harjani and Sadarangani established the following results.
Theorem 1.3 (Harjani and Sadarangani [3])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
T is continuous nondecreasing;
-
(iii)
there exists such that ;
-
(iv)
there exist such that for all with ,
Then T has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
Theorem 1.4 (Harjani and Sadarangani [3])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
if a nondecreasing sequence in X converges to some point , then for all n;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exist such that for all with ,
Then T has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
In [4], Amini-Harandi and Emami established the following results.
Theorem 1.5 (Amini-Harandi and Emami [4])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
T is continuous nondecreasing;
-
(iii)
there exists such that ;
-
(iv)
there exists such that for all with ,
Then T has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
Theorem 1.6 (Amini-Harandi and Emami [4])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
if a nondecreasing sequence in X converges to some point , then for all n;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists such that for all with ,
Then T has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
Remark 1.1 Jachymski [5] established that Theorem 1.5 (resp. Theorem 1.6) follows from Theorem 1.3 (resp. Theorem 1.4).
Remark 1.2 Theorems 1.3 and 1.4 hold if satisfies only the following conditions: φ is lower semi-continuous and (see, for example, [6]).
The following results are special cases of Theorem 2.2 in [7].
Theorem 1.7 (Ćirić et al. [7])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
T is continuous nondecreasing;
-
(iii)
there exists such that ;
-
(iv)
there exists a continuous function with for all such that for all with ,
Then T has a fixed point.
Theorem 1.8 (Ćirić et al. [7])
Let be an ordered set endowed with a metric d and be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
if a nondecreasing sequence in X converges to some point , then for all n;
-
(iii)
T is nondecreasing;
-
(iv)
there exists such that ;
-
(v)
there exists a continuous function with for all such that for all with ,
Then T has a fixed point.
Remark 1.3 Theorems 1.7 and 1.8 hold if we suppose that (see, for example, [8]).
Let X be a nonempty set and be a given mapping. We say that is a coupled fixed point of F if
In [9], Bhaskar and Lakshmikantham established some coupled fixed point theorems on ordered metric spaces and applied the obtained results to the study of existence and uniqueness of solutions to a class of periodic boundary value problems. The obtained results in [9] have been extended and generalized by many authors (see, for example, [8, 10–23]).
In this paper, we will prove that most of the coupled fixed point theorems are in fact immediate consequences of well-known fixed point theorems in the literature.
2 Main results
Let be a partially ordered set endowed with a metric d and be a given mapping. We endow the product set with the partial order:
Definition 2.1 F is said to have the mixed monotone property if is monotone nondecreasing in x and is monotone non-increasing in y, that is, for any ,
Let . It is easy to show that the mappings defined by
for all , are metrics on Y.
Now, define the mapping by
It is easy to show the following.
Lemma 2.1 The following properties hold:
-
(a)
is complete if and only if and are complete;
-
(b)
F has the mixed monotone property if and only if T is monotone nondecreasing with respect to ⪯2;
-
(c)
is a coupled fixed point of F if and only if is a fixed point of T.
2.1 Bhaskar and Lakshmikantham’s coupled fixed point results
We present the obtained results in [9] in the following theorem.
Theorem 2.1 (see Bhaskar and Lakshmikantham [9])
Let be a partially ordered set endowed with a metric d. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exists a constant such that for all with and ,
Then F has a coupled fixed point . Moreover, if for all there exists such that and , we have uniqueness of the coupled fixed point and .
We will prove the following result.
Theorem 2.2 Theorem 2.1 follows from Theorems 1.1 and 1.2.
Proof From (v), for all with and , we have
and
This implies that for all with and ,
that is,
for all with . From Lemma 2.1, since is complete, is also complete. Since F has the mixed monotone property, T is a nondecreasing mapping with respect to ⪯2. From (iv), we have . Now, if F is continuous, then T is continuous. In this case, applying Theorem 1.1, we get that T has a fixed point, which implies from Lemma 2.1 that F has a coupled fixed point. If conditions () and () are satisfied, then Y satisfies the following property: if a nondecreasing (with respect to ⪯2) sequence in Y converges to some point , then for all n. Applying Theorem 1.2, we get that T has a fixed point, which implies that F has a coupled fixed point. If, in addition, we suppose that for all there exists such that and , from the last part of Theorems 1.1 and 1.2, we obtain the uniqueness of the fixed point of T, which implies the uniqueness of the coupled fixed point of F. Now, let be a unique coupled fixed point of F. Since is also a coupled fixed point of F, we get . □
2.2 Harjani, López and Sadarangani’s coupled fixed point results
We present the results obtained in [16] in the following theorem.
Theorem 2.3 (see Harjani et al. [16])
Let be a partially ordered set endowed with a metric d. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exist such that for all with and ,
Then F has a coupled fixed point . Moreover, if for all there exists such that and , we have uniqueness of the coupled fixed point and .
We will prove the following result.
Theorem 2.4 Theorem 2.3 follows from Theorems 1.3 and 1.4.
Proof From (v), for all with and , we have
and
This implies (since ψ is nondecreasing) that for all with and ,
that is,
for all with . Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.3 (resp. Theorem 1.4). The rest of the proof is similar to the above proof. □
2.3 Lakshmikantham and Ćirić’s coupled fixed point results
In [8], putting (the identity mapping), we obtain the following result.
Theorem 2.5 (see Lakshmikantham and Ćirić’s [8])
Let be a partially ordered set endowed with a metric d. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exists such that for all with and ,
Then F has a coupled fixed point.
We will prove the following result.
Theorem 2.6 Theorem 2.5 follows from Theorems 1.7 and 1.8.
Proof From (v), for all with and , we have
and
This implies that for all with and ,
that is,
for all with . Here, is the metric on Y given by
Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.7 (resp. Theorem 1.8). Then T has a fixed point, which implies that F has a coupled fixed point. □
2.4 Luong and Thuan’s coupled fixed point results
Luong and Thuan [18] presented a coupled fixed point result involving an ICS mapping.
Definition 2.2 Let be a metric space. A mapping is said to be ICS if S is injective, continuous and has the property: for every sequence in X, if is convergent, then is also convergent.
We have the following result.
Lemma 2.2 Let be a metric space and be an ICS mapping. Then the mapping defined by
is a metric on X. Moreover, if is complete, then is also complete.
Proof Let us prove that is a metric on X. Let such that . This implies that . Since S is injective, we obtain that . Other properties of the metric can be easily checked. Now, suppose that is complete and let be a Cauchy sequence in the metric space . This implies that is Cauchy in . Since is complete, is convergent in to some point . Since S is an ICS mapping, is also convergent in to some point . On the other hand, the continuity of S implies the convergence of in to Sx. By the uniqueness of the limit in , we get that , which implies that as . Thus is a convergent sequence in . This proves that is complete. □
The obtained result in [18] is the following.
Theorem 2.7 (see Luong and Thuan [18])
Let be a partially ordered set endowed with a metric d. Let be an ICS mapping. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exists such that for all with and ,
Then F has a coupled fixed point.
We will prove the following result.
Theorem 2.8 Theorem 2.7 follows from Theorems 1.7 and 1.8.
Proof The condition (v) implies that for all with and ,
that is,
for all with , where is the metric (see Lemma 2.2) on Y defined by
Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.7 (resp. Theorem 1.8). Then T has a fixed point, which implies that F has a coupled fixed point. □
2.5 Berind’s coupled fixed point results
The following result was established in [11].
Theorem 2.9 (see Berinde [11])
Let be a partially ordered set endowed with a metric d. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exists a constant such that for all with and ,
Then F has a coupled fixed point . Moreover, if for all there exists such that and , we have uniqueness of the coupled fixed point and .
We have the following result.
Theorem 2.10 Theorem 2.9 follows from Theorems 1.1 and 1.2.
Proof From the condition (v), the mapping T satisfies
for all with . Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.1 (resp. Theorem 1.2). Then T has a fixed point, which implies that F has a coupled fixed point. The rest of the proof is similar to the above proofs. □
2.6 Rasouli and Bahrampour’s coupled fixed point results
Theorem 2.11 (see Rasouli and Bahrampour [20])
Let be a partially ordered set endowed with a metric d. Let be a given mapping. Suppose that the following conditions hold:
-
(i)
is complete;
-
(ii)
F has the mixed monotone property;
-
(iii)
F is continuous or X has the following properties:
-
() if a nondecreasing sequence in X converges to some point , then for all n,
-
() if a decreasing sequence in X converges to some point , then for all n;
-
-
(iv)
there exist such that and ;
-
(v)
there exists such that for all with and ,
Then F has a coupled fixed point . Moreover, if for all there exists such that and , we have uniqueness of the coupled fixed point and .
We have the following result.
Theorem 2.12 Theorem 2.11 follows from Theorems 1.5 and 1.6.
Proof From the condition (v), the mapping T satisfies
for all with . Thus we proved that the mapping T satisfies the condition (iv) (resp. (v)) of Theorem 1.5 (resp. Theorem 1.6). Then T has a fixed point, which implies that F has a coupled fixed point. The rest of the proof is similar to the above proofs. □
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This work is supported by the Research Center, College of Science, King Saud University.
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Samet, B., Karapınar, E., Aydi, H. et al. Discussion on some coupled fixed point theorems. Fixed Point Theory Appl 2013, 50 (2013). https://doi.org/10.1186/1687-1812-2013-50
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DOI: https://doi.org/10.1186/1687-1812-2013-50